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Journal of Applied Mathematics and Stochastic Analysis, 11:1 (1998), 59-71.

THE STATIONARY G/G/s QUEUE

PIERRE LE GALLFrance Telecom, CNET

Parc de la BrengreF-92210 Saint-Cloud, France

(Received November, 1997; Revised February, 1998)

The distribution of the queueing delay in the stationary G/G/s queue isgiven with an application to the GI/G/s queue and to the M/G/s queue.

Key words: G/G/s Queue, GI/G/s Queue, First Come-First Served,Factorization, Singular Points.

AMS subject classifications: 60K25, 90B22.

1. Introduction

In this paper we evaluate the distribution of the queueing delay in the stationaryG/G/s queue. The particular case of the GI/G/s queue was extensively studied ear-lier by Pollaczek [3]: it proves difficult to write down the equations for the GI/G/squeue, whose partial solution can only be derived after long and complex calculationsinvolving multiple contour integrals in a multi-dimensional complex plane.

In Section 2, we state the underlying assumptions and introduce notation beforeevaluating all singularities of the Laplace-Stieltjes transform of this distribution forthe GI/G/s queue for the limited case of the stationary regime. The method is thenextended to the case of the G/G/s queue (Section 3).

In Section 4, we derive the constraints to be satisfied in order that this Laplace-Stieltjes transform be holomorphic. To do so we propose a factorization method,which is more general than the Wiener-Hopf type decomposition.

In Section 5, we derive an expression for the distribution of the queueing delay inthe stationary G/G/s queue together with its asymptotic behavior for long delays. InSection 6, we apply our results to the case of the GI/G/s queue. Finally, in Section7, we consider the M/G/s queue.

The method presented requires relatively simple calculations making it possible toconsider the evaluation of local queueing delays in multiserver queueing networks.

Printed in the U.S.A. (C)1998 by North Atlantic Science Publishing Company 59

60 PIERRE LE GALL

2. Notation, Assumptions and Preliminary Results

2.1 The Stationary G/G/s Queue

We consider a queue handled by a multiserver of s identical servers.

a) The Arrival Process: We assume a metrically transitive, strictly stationary pro-cess of successive, non-negative interarrival times. Let N(t) denote the random num-

ber of arrivals in the interval (0, t]. We write dN(t)--1 or 0 dependent on whetheror not there is an arrival in the infinitesimal time interval (t, t + dr). We exclude thepossibility of simultaneous arrivals. We can then write:

E[dN(to) dN(to + t)] E[dN(to) p(t)dt, (1)

where p(t) is the arrival rate at time (t + to) if an arbitrary arrival occurred at timeto We let, for Re(z < 0:

ezt. p(t)dt Cl(Z 90, x(Z), (2)0 x=l

where po, x(z) corresponds to the xth arrival following the epoch t0. However, the sta-tionary assumption and Abelian theorem gives: Limz_0z- al(Z A, where A is themean arrival rate. In a more general way, we may write, for j- 1,2,..

E[dN(to) dN(to + tl)...dN(to + tI +... + tj)][EdN(to)]. fj(tl...tj), dtl...dtj, (3)

and for R(zj) < O, j 1,2,..

ezl .dtl... e 3 3. dtj.fj(tl. ..tj) aj(Zl...zj). (4)0 0

In the case of a renewal process, the successive arrival intervals Yn are mutually in-

dependent and identically distributed and we let" o(Z)- NezYn for R(z)< O. Ex-pression (2) becomes: 0(z) (5)Ol(Z) 1 o(Z)’and (4) becomes:

Oj(Zl...Zj) Ol(Zl)...OZl(Zj). (6)

b) The Service Times: The successive service times Tn are mutually independentand independent of the arrival process. They are identically distributed in accordance

with a distribution function F(t) and we let (z)- EeT’, for Re(z < O. Weexclude the possibility of bulk service; consequently:

rl(0) FI( + 0) 0. (7)

c) The Service Discipline: The service discipline is "first come-first served’, andthe s servers are indistinguishable.

The Cotationary G/G/s Queue 61

d) Traffic Intensity:one:

The traffic intensity (per server) is supposed to be less than

[EdN(t)]’[E(Tn)]r- s <1. (8)

Under this condition, Loynes [1] demonstrated the existence of the stationaryregime.

e) Queueing Delay: Let ’n denote the queueing delay of the nth customer, and -of an arbitrary customer. Note: Since the term "waiting time" means "sojourn ime"in Little’s formula, for clarity we prefer to use the term "queueing delay" for thequeueing process only.

f) Contour Integrals: In this paper, we use (Cauchy) contour integrals along theimaginary axis in the complex plane. If the contour (followed by the bottom to thetop) is to the right of the imaginary axis (the contour is closed at infinity to theright), we write f. If the contour is to the left of the imaginary axis, we write f.

+o -oUnless it is necessary to specify whether the contour is to the right or to the left ofthe imaginary axis, we write f.

0

2.2 Preliminary Results (GI/G/1 Queue)

It will be useful to refer to Pollaczek [2] for the queue GI/G/1. For Re(q)>_ O, wehave"

_ml"f0{q -t-1 1 1Ee qr Exp{1 } log N0(I dl},

with" No(l 1 0( 1)" 1(1)"(9)

Note that:

1 1 1Exp{-zl" [q -t- 1 1 ]" lOg(1 (o( 1))" dl } 1, (9a)+o

since we have Io(- 1) < 1 for R.e(l) > 0, and, consequently, there are no singu-lar points in this region. We then multiply (9a) by (9), which leads to the substitu-tion" N0(l)N(I)--NI(I) 1 o( 1)

1 Cl( 1)" [91(1) 1], (10)

where c(- 1)is defined by (5).be written, for Re(q) >_ 0, as

Finally, the functional Ee -qr, defined by (9), may

with:

0 q+l

N1(1 --1-[1" Ctl(- 1)]" (1( 1.)

1 }" log NI(I) dl],1 (11)1 Ctl( 1)" [1(1) 1].

To find the singular points of this expression, we cross the poles 1- 0 and1---q in the integrand, and have (I)- (q) denote new Expression (11), forRe(l) < 0 and Re(l) < -q. We get the following Wiener-Hopf type of decomposi-

62 PIERRE LE GALL

tion for Re(q) < O, where -(q)is holomorphic for Pe(q) < O, and

G-(q)(I) + (q) (1 ).N1 q). (12)

Here, r/ denotes the traffic intensity and the probability of the delay. The roots ofthe denominator define the singular points of Ee-qr. Conversely, from (12) we maydeduce Expression (11). We intend to extend this procedure to the case s > 1 by ini-tially defining the singular points, and then defining the kind of decomposition.

3. The Singular Points [for Re(q) < 01From Expressions (10) and (12), the singular points of Ee -q" in the case of the sta-tionary GI/G/1 queue are also the singular points of

1 1 990(q)[1 990(q)]" E [99o(q)]n" [991 q)]n.NI( q) 1 990(q) 991( q) n 0

(13)

The latter may be rewritten as:

1 1 1 / dz1 q 1 99o(q)NI(-q) 27ri Zl q + z 1 -99o(q)" 991(Zl)" (14)

+0

In the stationary regime, the distribution of the queueing delay is independent of theinitial conditions. For instance, in the case of a large number of arrivals prior to

ntime zero, the busy period is then very long. The terms (W,x- V,x) and [p0(q)]n.

[991(- q)]n serve to evaluate the queueing delay of the nth customer following the ear-lier arrivals.

With the same initial conditions as before, the only change for the GI/G/s queueis to note that the sequence n of successive terminations of service times is nowdefined by s strings j of successive service times T,x(j), j- 1,...,s in parallel and bythe expression

n n

T,Xl( E (s)] nI +... + ns n, (15)n MiniE 1),..., T,XsA 0 As

0

where all possible groups (nl...ns) are considered.use the following expression given by Pollaczek [3]"

To evaluate this sequence we will

1 dZl dzsexp[- q. min + (al...as) 1 (2ri)S+J0-Yi-1...+jwith: Re(q + z,) > O,

u=l

q exp( au zu),q+ zv v=l

t=l(16)

where: min + (a1. ..as) Max[0, min(al...as) ].GI/G/s queue, (14)becomes:

Consequently, for the stationary

The Stationary G/G/s Queue 63

1 =1_ 1 /dZl ]’dzsNs( q) (2ri)--" -fi-1""" zs+o +o

q I 1 -9%(q)

q+ zv j= ll --(-’(zj)

which rnay be written, due to (5), as

Ns( q) (2ri)------" -i--1" zs

+o +o q+ zv j=ll--Ctl(q)’[l(Zj)-1]"--1

(17)

(18)

Note that the transformation from (14) to (18) depends on the arrival processthrough [OZl(q)]nl...[Ol(q)]ns. For stationary G/G/s queue (with the same initial con-

ditions) we have to make the following substitution, due to (6)"

[1 (q)]nl’" "[Ctl(q)]ns--*OZn(q’" "q)’ rtI +... -t- rts n.

More simply, it may be noted that (11) depends on {1/N1(1)} through-logNl(l --log[1/Nl(l)]. We will see that this is also true for the stationaryG/G/s queue [see (33) below]. Consequently, it is sufficient to consider the followingsubstitution"

s-1

--n,s(Z1. ..Zs;q)- 1 + , (-1 "as- ,x(q"" "q)" 1

[(fll(Zj)- 1]

(19)

[g91(zj) 1].

This substitution is very simple. It is due also to a typical property of the arrivalpoint processes (with non-simultaneous arrivals); the logarithm of the characteristicfunctional with respect to N(t) allows to replace all the higher moments (3) by a

single integration of E[N(t)]. Finally, we can state:Theorem 1: (Singular points) For the stationary G/G/s queue, the singular points

of Ee- qr, for Re(q) < O, are those of the function:

with:

dZl / dzs q 1 (20)Gs(q) 1 (21i)------- -57-1... z-" Rs(Zl...Zs q)’+o +o q + 2_, zv

and:

s-1

R,s(Zl. .Zs; q) 1 + ,, 1 "as- ix(q"" "q)" l [l(Zj)- 1],j=+l

R,e(q + zu) > O,v-----1

where cs ,x(q’. "q) is defined by (4) for Re(q) < O"

/ eqt,k + 1. dta + l’"f eqts’dts’fs-x(tA+l" "ts)"o o

64 PIERRE LE GALL

Corollary 1: For the stationary GI/G/s queue, (6) gives OZs_ A(q’"q)- [Cl(q)]s- "and (18) gives:

/ dzl / dzs q l 1 (20a)1)s -i--1" z---"Gs(q)- 1(2ri

+0 "’+0 q+ z j=ll Cl(q) [l(Zj)-- 1],=1

4. The Factorization

We want to get a holomorphic function for Re(q) >_0, with singular points(le(q) < 0) as defined by (20). We will be obliged to introduce some auxiliary com-

plex variables z (i 1...s). Finally, we want to define a function Us(zl...zs; q), holo-morphic for Re(zi) >_ 0 (i.e., _> -5), i= 1...s, and Re(q) >_ 0, such as the singularpoints of Us(0...0; q) are dCnd by (20).

Consider the following integral, for I{.e(q + z) > O"u=l

[ Us(z1. .zs;q). (21)dzs qI(q)

+o +o q + zu

With our assumption for Us, the integrand is holomorphic for R(zj)> 0, j- 1...s.Therefore, we have: I(q) O. But, if we cross over poles zj 0 (j 1...s) from theleft hand side of the right-hand side, the residue of the integrand is" (-1)s

Us(0...0;q). We therefore deduce that

_(-1)s [dZl [dzs qUs(0. "0 q Z1 Zs;q) ()(. ]...]-o -o q + zu

In this integral, q is not a variable; rather it is only a parameter.following factorization"

Ftorization: We set1) (q...)" [l(Z) 11

s(Zl "zs; q) j 1

Introduce the

H Mi(Zl’" "Zs; q)’ (23)Rs(Zl""Zs’q) i= 1

where Rs is defined by (20) and Us is holomorphic for Re(zi)>_ 0 (i 1...s) andRe(q) >_ O, with the following conditions for Mi:

a) M is holomorphic for Re(zi) < O, 1...s;i-1

b) Mi(Zl...Zi_l, -q- zu, zi+l...Zs;q)- 1.t--1

Now, we want to evaluate the integral (22) in the region FCe(zi)< 0, i= 1...s. Inthis region, the equation P%(zl...zs;q)=0 has no root because of the inequalityl(Zj) < 1; thus the product

(-- 1)s- [l(Zj)- 1],j=+l

in (20), always has a positive real part. Consequently, it is the same for Ks.When we integrate in complex plane zs, Ms has no singularities for P(zs) < 0 due

The Cotaionary G/G/s Queue 65

s-1to condition (a). In this plane, we find only the pole: z -q- z,. To evaluate

u=lthe residue, we have to apply condition (b): as a result, M disappears. This will bethe same for the integration in successive planes zs -,...,z1. Finally, (22) becomes:

c%(q...q)" -I [9l(Zj)-- 1]Us(O" .O; q) l J dZl J dzs q j=l

(27ri)s -1"’" z---ft" l{s(Zl" q)+0 +0 q-Jr- zv "zs’

/2=1

where zj 0 (j- 1...s) is not a pole. To simplify the integrand, firstly we considerthe GI/G/s queue. We write, due to the symmetry with respect to variables zj:

fl [(zj)- 1]=1 I( Ctl(a)’[(fll(Z)--l] )c(q...q) j

R(Zl...z,q j =11 i) [-1i 1]

(1- 1(- 1)s

"j=l 1-Cl(q). [(zy)- 1]

(-1)s. [1 +,(). (-1)s-’x 1 .].,= II [1 Ol(q)’[991(zj) 1]]

j=,k+l

For the G/G/s queue, we may apply the same reasoning which led to substitution(19) and used the following substitution, corresponding to formula (6)" [ch(q)]S-’Xcs ;(q...q). Under the integrand we may write:

Cts(q’"q)" -I [l(Zj)-- 1]2=1

Is(Zl. .Zs; q

s--1

=(-1) .[1+ , .(-1 1P%- ,k(z,k + 1"" "Zs; q)]"

Between brackets, term 1 and terms ,( > 0), where at least one zj is missing, do notcontribute to the integral [for Re(zj)> 0]. The expression of Us(0...0;q becomeswith the only term ,- 0"

U(O..0; q) 1(i)

Let

dZl Jdzs q 1 (24)-1" Z-----" R,s(Zl" .Zs; q)"+o +o q + zv

Vs(Zl...Zs; q) 1 Us(Zl. .Zs; q). (25)

We can finally write, for Re(q) >_ O"

v(o...o; q) a(q), (26)

where Gs(q) is Expression (20) giving the singular points of Ee- qr. To summarize:

66 PIERRE LE GALL

The holomorphic function Vs(Zl...zs;q) defined by (25) and by factorization, in(23), with condilions (a) and (b), leads to (26) giving the singular points of Ee -qr

for the slalionary G/G/s queue.Note: 1) For s 1, the factorization, in (23), is of the Wiener-Hopf type:

_UI(Zl;q /l(Zl; q) 1 Ml(Zl;q)Rl(zl;q)il(zl;q)-(i"2) For s > 1, it is not sufficient to define Gs(q) and the singular points; to define

Ee -qr, it is also necessary to use the factorization, in (23), different from theWiener-Hopf type.

5. The Queueing Delay (Stationary G/G/s Queue)

5.1 The Distribution

We introduce the following holomorphic function for Re(zj)>_ 0 (j-1...s) andR(q) >_ 0"

Vs(Zl...Zs; q) 1 Us(Zl...Zs; q)

1 Exp[(2lia [q q- 1 -f-Zl 1

-0

1 1 S.]dl"" s 1 + "ds" lgNs(ffl"" "ffs)-o q+ z,+(s Zs-

v--1

(27)

i-1with Re(q + E z, + i) > O, i- 1...s, and

1 As(z1"" "Zs)Ns(Zl. .Zs)= Bs(Zl. .Zs) (28)

where: As(z1...zs) 1)s. { 1-] ; l[(fll(Zj)- 1]}. cs( Zl, z z2,... Z,),s-1

’=1

Bs(Zl...Zs)- 1 + A (- 1)

j=A+I

and j(Zl...zj)is defined by (4).Now, we decompose Us for Re(zi)< O, i- 1...s. First, consider variable zs and

complex plane s in (27). We go to the region Re(s)> Re(zs) and, consequently, wecross pole s- zs and set

Us(Zl...Zs; q) Hs(Zl...Zs; q)" Ms(Zl...Zs; q), (29)

where the integrand of Hs is the residue of the integrand of Us(z1...Zs; q). We have:


Hs(Zl...Zs); q) Exp[ 1(27ri)s-1

1 .,.+ .1 d1

-0 (30)

1

-o q+ Z+s- 1,=1

1+Zs (s 1

]" ds 1" lgNs(l""s 1, Zs)]"

Ms(Zl...zs;q) is still defined by (27) but with R(s)> R(zs). Ms is therefore holo-morphic for Re(zs)< 0. Proceeding in the same way for variable zs_ 1 and planes- 1 in (30), we set as above

Hs(Zl" "zs; q) Hs- l(Zl "’’zs;q)" Ms- l(Zl "’’zs;q)’ (31)

where the integrand of Hs_ 1 is the residue of that of H,. We therefore have"

Hs_ l(Zl...Zs;q)- Exp[ 1(27ri)s-1

1 1[q / 1 /Zl 1]" dl" (32)

-0

1

-0 q+ }2 z.+_u=l

1 ]" ds 2" lgNs(l’" "s 2, zs 1, Zs).Zs 2 s 2

Ms- 1 is defined by (30), with R(s_ 1) > t(Zs- 1)" Ms- is holomorphic forRe(Zs_l) < 0. Continuing in this way, we derive the decomposition, in (23) withj- 1...s"

-1/ 1Mj(Zl""zs;q) ExP[(27ri)J [q / (i /-0

1

-o q/ zuW(j

1 ]. dj. logNs(l...j, zj + 1"" "Zs)]"Zj--j

(33)

Mj satisfies conditions (a) and (b); finally Us satisfies the factorization, in (23), andconsequently, Vs satisfies (25). Moreover, from (25), we have Vs(0...0;0 1. Wewill consider specially the function (Ee -qr) relating to the queueing delay r ofdelayed customers, defined by Vs(O...O;q (I-P)+P.Ee -qr, where P is theprobability of delay corresponding to ]ql increasing indefinitely. Consequently, wemay drop the terms [1/(zj-j)] in integrand (27). Since we know that the solutionis unique, we can state:

Theorem 2: (G/G/s Queue) For the stationary G/G/s queue, the queueing delay7 of an arbitrary delayed customer is given, for Re(q) > O, by:

Ee qr 1 EzP{(2ris J dl i d(nus(: lflgs(l " )} (34)q+l q s

where Ns((1...(s is defined by the long Expression (28).

68 PIERRE LE GALL

5.2 The Asymptotic Distribution

The asymptotic distribution corresponds to the (real) singularity closest to the origin.We wish to evaluate it. To more readily evaluate this singular point, we transform(20) by noting a certain symmetry with respect to variables zj close to this realpoint. We successively set:

zj- l-g.(-q+j), j-1...s; (35)

(Z)- 1()" (36)

We deduce, for Re(-q + 4j) > 0 (j- 1...s)"

Gs(q) 1 (21i)s q + 1"" q -1- s+o +o

withs--1

R’s(l""s;q)--1+o()’(-: 1)s-’ "as- )(q"" "q)" 1=,+

[p(- q + ffj)- 1].

(37)

We go now to the region Re(-q + j)< 0poles 1 ="" =s -0 and 1 =’"=s--q"Re(-q+ffj)<0 (j-l...s):

Gs(q (_@)s- 1. 1n;(0...0,q)

(j- 1...s)and, consequently, cross

Expression (37) becomes, for

(38)s f dl / d______s q______. 1

(27ri)s q + 1"" q -t- s p /’s(l"" "s, q)"

For Re(-q+j)<0, we have 19(-q+j) <1. It follows that the real partof [p( q + (jj 1] is positive, and it is the same for R;((1...,; q), R(q) > 0. Itdoes not appear that any singularity exists in the integrand of (38) for Re(j < 0(j- 1...s). The integral is equal to zero, and we deduce for our transformation thataround the wanted singular point,

Gs(q (_@)s- 1. 1 (39)n;(0...0, q)"

Rule: Finally, the singular point (for the asymptotic expression of the queueingdelay distribution) is the real root qo (closest to the origin) of the following equation,for Re(q) < O"

s--1

/;(0...0;q) 1 + A "cs-’x(q’"q)’[1 -7(- q)] -a 0, (40)

where 9(- q)is defined by (36).

The tationary G/G/s Queue 69

6. The Stationary GI/G/s Queue

To get Ee- qr, we apply Theorem 2 where (28) becomes, due to (6):

1 As(Zl" "zs)Ns(Zl...zs) Bs(Zl. .zs)’

withs

As(Zl’"Zs) (- 1)s" H {[I(ZJ 1]. al(- Ej--1 =1

s-1 JBs(Zl...zs)- 1 + I (- 1 {[Tl(zj)- 1]. Ctl(- Z,)}.

j=A+I ,=1

(41)

To get the asymptotic distribution, equation (40) becomes"

R;(0...0; q) [1 Cl(q). [99( q) 111s1 -o(q)’P(-q)

1 o(q)This leads to the equation"

(42)

1 Po(q)" 9i( q) 0. (43)

The asymptotic expression of the queueing delay distribution corresponds to the realroot qo (closest to the origin) of (43). We may conclude:

Rule for the asymptotic distribution: The arbitrary delayed customer of the sta-tionary GI/G/s queue has the following queueing delay asymptotic distribution"

F(t,s)-F(st, 1), (44)

where F(t, 1) is the queueing delay distribution in the GI/G/1 queue corresponding tothe couple 0(z) and 1()" But, to evaluate the probability of delay, we have to usethe factorization, in (23) for s > 1 rather than the Wiener-Hopf decomposition(corresponding to s 1).

Note: 1) For the GI/D/s queue, the cyclic nature of service time terminationsleads to a factorization, similar to the Wiener-Hopf decomposition, but for the couple[7)0(z)]s and 7)l(Z), instead.

2) For the GI/M/s queue, there is only one singular point. Consequently, (44) isvalid for any value of > 0).

7. The Stationary M/G/s Queue

In (41), the function Cl(Zi) becomes, in case of Poisson arrivals with A for the arrivalrate:

A (45)o1 (zi) zi.

Expression (4.1) becomes:

70 PIERRE LE GALL

with

1 As(Zl...zs)Ns(Zl...zs) Bs(Zl. .zs)’ (46)

Consider the term for j A + 1 by writing it as

zA+I z1 --.. -k- z,k _F 1"Due to the symmetry in (34) and (46), we do not change anything for the value ofintegrals in (27) by writing successively, for zj with j _< A + 1:

A.{I(ZA+I) -1 zj } 1 1A.{991(z,+1)-1} Zl+’"+ZA+l-*, - ZA -- 1 "Zl +""" - ZA -t- 1"ZA -t- 1 Zl -"’5c ZA + 1

If we apply this transformation to the successive terms in (46), Theorem 2 allows usto state:

Theorem 3: (M/G/s Queue) For the stationary M/a/s queue, the functionEe-qr relating to the queueing delay 7 of an arbitrary delayed customer is 9iven by

I As(Zl""zs, (4)Ns(Zl...zs) Bs(Zl. .zs

with

As(z1 z) -(-1)s I{A’[pl(Zj)-1]}zj=l

_l)-,X IB(Zl...z) 1 + (i- A)IA=O "j=A+I{A.l(Zj) -1}zj

8. Conclusion

We may note a discrepancy in the case of the GI/M/s queue. We recall that a keypoint in this paper relating to (15) is the one that defines the sequence tn of success-ive terminations of service times during the congestion period; and it is not a Markovprocess. The traditional Markovian assumption was sufficient to evaluate the eventsfor a single arrival, and for the asymptotic expression of the queueing delaydistribution of a delayed customer. However, to evaluate the probability of the delayand occupancy probabilities it is necessary to properly evaluate the correlationsbetween two successive arrivals by taking into account the starting epoch and age (innumber of service times) of a busy (quite congested) period. It is also imperative tointroduce the sequence tn defined in (15), and modify the basic Poisson process toevaluate the service time terminations during congestion. Surprisingly, for a verylong time, the discrepancy between the formulas presented in Pollaczek [3] andTakcs formula (for the GI/M/s queue) has never been stressed.


References

[1]

[2]

Loynes, R.M., The stability of a queue with nonindependent interarrival and ser-vice times, Proc. Cambridge Philos. Soc. 58 (1962), 494-520.Pollaczek, F., Problmes stochastiques poss par le phnomne de formationd’une queue d’attente un guichet et par des phnomnes apparents, Mmorialdes Sciences Mathmatiques, Gauthier-Villars, Paris CXXXVI (1957).(= GI/G/1 queue; in French).Pollaczek, F., Thorie analytique des problmes stochastiques relatifs un

groupe de lignes tlphoniques avec dispositif d’attente, Mmorial des SciencesMathmatiques, Gauthier-Villars, Paris CL (1961). (=GI/G/s queue; inFrench).

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THE STATIONARY G/G/s QUEUEdownloads.hindawi.com › journals › ijsa › 1998 › 587917.pdf · The Cotationary G/G/s Queue 61 d) Traffic Intensity: one: The traffic intensity (per